Discrete group actions on 3-manifolds and embeddable Cayley complexes
Discrete group actions on 3-manifolds and embeddable Cayley complexes
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发表时间：
2022-08
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通讯作者：
Agelos Georgakopoulos;George Kontogeorgiou
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作者：
Agelos Georgakopoulos;George Kontogeorgiou
我们证明了一个群$\Gamma$允许一个离散拓扑（等价地，光滑）作用在某个单连通3-流形上当且仅当$\Gamma$有一个Cayley复形可嵌入-在某些自然限制下-在以下四个3-流形之一中：（i）$\martbb {S}^3$，（ii）$\martbb {R}^3$，（iii）$\martbb {S}^2 \times \martbb {R}$，（iv）$\martbb {S}^3$中驯服康托集的补集。
We prove that a group $\Gamma$ admits a discrete topological (equivalently, smooth) action on some simply-connected 3-manifold if and only if $\Gamma$ has a Cayley complex embeddable -- with certain natural restrictions -- in one of the following four 3-manifolds: (i) $\mathbb{S}^3$, (ii) $\mathbb{R}^3$, (iii) $\mathbb{S}^2 \times \mathbb{R}$, (iv) the complement of a tame Cantor set in $\mathbb{S}^3$.